A topological space 
such that for every closed subset 
of 
and every point 
, there is a continuous
function ![f:X->[0,1]](/images/equations/CompletelyRegularSpace/Inline5.svg)
such that 
and 
.
This is the definition given by most authors (Kelley 1955, p. 117; Willard 1970, pp. 94-95). However, some authors (e.g., Cullen 1968, p. 130) require the
additional condition that 
be a T1-space. In any case, every
completely regular space is regular, and the converse is not true.
